3.1. Convection patterns

Figure 2 presents the flow patterns at the measured fluid height with $\varOmega = 1.12 \times 10^4$ and $\varepsilon = 0.49$. When a flat bottom plate is used (i.e. the reference cell without external forcing), the flow fields of the vertical vorticity $\omega (\boldsymbol {r})$ are characterized by columnar vortices, which exhibit stochastic horizontal motions as reported in previous studies (Chong et al. Reference Chong, Shi, Ding, Ding, Lu, Zhong and Xia2020; Ding et al. Reference Ding, Chong, Shi, Ding, Lu, Xia and Zhong2021). Despite their random motion, the vortices maintain approximately a constant distance $\lambda _0 = 13.75{\pm }1.57$ mm from their neighbouring vortices (figure 2a). In the spatial Fourier spectrum $F(\boldsymbol {k})$ of the vorticity field calculated in the central region (figure 2d), a crater-like structure with radius $k_0 = 2{{\rm \pi} }/\lambda _0 (457.1{\pm }52.4$ m$^{-1}$) is apparent, indicating that the vortices are distributed with random orientations but with a preferred spacing. When periodic topographic structures are constructed on the bottom plate, they modulate both the local temperature and the shearing interaction of the fluid with the solid surface, leading to new convection patterns. Figures 2b and 2c show the vorticity fields when the bottom surface is textured with a square and a hexagonal array of cylinders, respectively, with their spacing $\lambda$ chosen close to the intrinsic wavelength of the vorticity field $\lambda _0$. Since the fluid overlying the raised cylinders is relatively hotter than the background fluid at the same fluid height, upwelling vortices, i.e. cyclones when observed in the lower half of the fluid layer (see e.g. Sakai (Reference Sakai1997), King & Aurnou (Reference King and Aurnou2012), for illustrations), tend to form above the cylinders, forming a 1 : 1 commensurate structure with respect to the bottom texture. The downwelling vortices (anticyclones), however, appear in between the raised cylinders. In the square-patterned cell (figure 2b), both the cyclones and anticyclones constitute a regular square lattice. The flow pattern induced by a hexagonal array of cylinders (figure 2c), however, consists of a hexagonal lattice of anticyclones with a cyclone located at the hexagon centre. Such patterns are stationary and persist during the experiment. Figures 2e and 2f present the Fourier spectra of the vorticity fields in the central region of figures 2b and 2c, respectively. In these spectra we see clear peaks located precisely at the wave vectors $\boldsymbol {k}_{f} = \boldsymbol {k}_i$ ($i = 1,2\ldots$) of the periodically imposed textures. These peaks of $F(\boldsymbol {k})$ are all sharp and their amplitudes are approximately equal, suggesting that regular patterns with prescribed periodicity and symmetry are developed. Near the sidewall region ($r\,{\ge }\,100$ mm) where the imposed texture is absent, the flow field is time-varying and the vortex dynamics is largely influenced by the retrograde travelling plumes within the region of the boundary zonal flow (de Wit et al. Reference de Wit, Aguirre Guzmán, Madonia, Cheng, Clercx and Kunnen2020; Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2020).

3.2. General features of the patterns in the $k_f - {\varepsilon }$ phase diagram

The observed spatial pattern can be quantified by the radial distribution function $g(r)$ of the vortices, which is defined as the ratio of the actual number of cyclones lying within an annulus region of $r$ and $r + {\delta }r$, to the expected number for uniform vortex distribution (Chong et al. Reference Chong, Shi, Ding, Ding, Lu, Zhong and Xia2020). Figure 3(a) shows $g(r)$ for cyclonic distribution for various $\varepsilon$ in the square-patterned cell with $\lambda = 14.14$ mm. Near onset ($\varepsilon = 0.09$), multiple sharp peaks appear in $g(r)$, which are located at distances that match the main and subharmonic wavelengths $r_{ij}$ of the forced square pattern at the bottom plate, fulfilling the condition $r_{ij} = {\lambda }\sqrt {i^2 + j^2}$ (for $i,j = 0,1,2\ldots$ and $i + j\,{\ge }\,1$). With increasing $\varepsilon$, the peak amplitudes in $g(r)$ decrease while the peak widths increase, signifying a less regular flow pattern. The multiple-peak structure is eventually flattened, and $g(r)$ becomes close to a uniform distribution for $\varepsilon \,{\ge }\,4.0$, where we see the vortices exhibit apparent horizontal motions. We examine as well the role of the periodicity of external forcing on the convection pattern. Figure 3(b) presents results for $g(r)$ near onset ($\varepsilon = 0.49$) for various cylinder spacings $\lambda = 10.00, 14.14, 20.00, 28.28$ mm. We see that multiple peaks still appear at the main and subharmonic wavelengths $r_{ij}(\lambda )$. For $\lambda \,{\ge }\,20.00$ mm, the first peak is found near $r = \lambda _0$, which is associated with the intrinsic wavelength of the flow field. When $\lambda$ is far from $\lambda _0$, the maxima of $g(r)$ become less dominant and $g(r)$ approaches the result of the reference cell.

Figure 3. (a,b) Radial distribution function $g(r)$ of cyclones in the square-patterned cell. Results are for various $\varepsilon$ with a constant cylinder spacing $\lambda = 14.14$ mm in (a), and for various $\lambda$ with $\varepsilon = 0.49$ in (b). The red, blue and green arrows in (b) denote the main and subharmonic wavelengths $r_{ij}/{\lambda _0}$ of the imposed pattern for $\lambda = 14.14, 20.00$ and $28.28$ mm, respectively. The dotted line in (b) shows the results for the reference cell. (c) Contour plot of the cross-correlation coefficient $C$ of the vorticity field and bottom texture in the $k_f/k_0 - \varepsilon$ phase diagram. Open symbols are data points measured in the square-patterned cell; colour contours are estimated from interpolation between these points. Results are for $\varOmega = 1.12 \times 10^4$.

The degree of matching between the flow pattern and the bottom texture can be evaluated through the cross-correlation coefficient $C$ of the vorticity field and the bottom texture, defined as $C = {\langle }(\omega (\boldsymbol {r}) - \bar {\omega })(M(\boldsymbol {r}) - \bar {M}){\rangle }/\sqrt {{\langle }(\omega (\boldsymbol {r}) - \bar {\omega })^2{\rangle }{\langle }(M(\boldsymbol {r}) - \bar {M})^2{\rangle }}$, with $M(\boldsymbol {r}) = 0$ (or $- 1$) for the flat (or raised) area representing the bottom surface profile. Angle brackets ${\langle }{{\,\cdot\, }}{\rangle }$ denote a spatial average. Figure 3(c) summarizes the results for $C$ in the square-patterned cell for varying $\varepsilon$ and wave vector $k_{f} = {\vert }\boldsymbol {k}_1{\vert } = {\vert }\boldsymbol {k}_2{\vert }$ of the external forcing. In this phase diagram, we see that $C(k_{f},\varepsilon )$ has a single maximum ($C\,{\approx }\,0.7$) occurring at ($k_{m} = 0.97k_0, \varepsilon _{m} = 0.09$), which implies the optimal conditions for pattern selection. In the vicinity of $(k_{m}, \varepsilon _{m})$, the spatial distribution of the vortices closely conforms to the bottom texture. The coefficient $C$ decreases if the control parameters $(k_{f}, \varepsilon )$ deviate from $(k_{m}, \varepsilon _{m})$. The decreasing rate of $C$ with increasing $\varepsilon$ is lowest when a near-resonant external forcing $(k_{f}\,{\approx }\,k_0)$ is chosen. The convection pattern and the imposed texture become essentially uncorrelated (with $C \le 0.1$) when $(k_{f}, \varepsilon )$ are set apart from $(k_{m}, \varepsilon _{m})$. Interestingly, when $k_{f}\,{\approx }\,k_{m}$, $C$ remains well above zero for $\varepsilon < 0$, suggesting that under external forcing, convection sets in with finite amplitude in the subcritical regime.

3.3. Variations of the vorticity field near onset

We measure the time-averaged vorticity modulus ${\langle }{\vert }{\omega }{\vert }{\rangle }$ in an area of $65.8 \times 54.8$ mm$^2$ at the centre of the cell, while slowly scanning ${{\rm \Delta} }T$ in the near-onset range $-0.4 \le \varepsilon \le 0.4$. Results for ${\langle }{\vert }{\omega }{\vert }{\rangle }(\varepsilon )$ for the reference cell and two forced cells are shown in figure 4. Overall, these data suggest two distinct types of bifurcations when $\varepsilon$ increases from below and crosses zero. The reference cell data reveal a sharp transition from a non-convection state with ${\langle }{\vert }{\omega }{\vert }{\rangle } = 0$ for $\varepsilon \le 0$, to a convection state in which ${\langle }{\vert }{\omega }{\vert }{\rangle }$ increases rapidly for $\varepsilon \,{>}\,0$. For both forced cells, however, ${\langle }{\vert }{\omega }{\vert }{\rangle }$ remains positive for $\varepsilon \,{\ge }-\!0.4$ and grows relatively slowly with increasing $\varepsilon$, suggesting a smooth transition. The three inset panels in figure 4 present the vorticity fields captured in the three cells with approximately the same subcriticality ($\varepsilon \,{\approx }-\!0.1$). They demonstrate that while the fluid is still quiescent in the reference cell, apparent square and hexagonal lattices of convective vortices have formed in the forced cells.

Figure 4. Bifurcation curves of ${\langle }{\vert }{\omega }{\vert }{\rangle }(\varepsilon )$ near onset. The circles represent the data of the reference cell. The squares and triangles represent respectively data for the square-patterned cell with $\lambda = 14.14$ mm and the hexagon-patterned cell with $\lambda = 17.32$ mm. Error bars denote the standard deviation. The dotted line shows the fitted square-root law for the reference cell, and the solid lines are the predicted imperfect bifurcation curves for the forced cells. Inset panels: time-averaged vorticity fields of the three cells for $\varepsilon \approx {-}0.1$. Data are for $\varOmega = 1.12 \times 10^4$.

3.4. A theoretical model of the bifurcation dynamics

In an effort to understand the $\varepsilon$-dependence of ${\langle }{\vert }{\omega }{\vert }{\rangle }$ near onset, we propose a phenomenological Ginzburg–Landau-like model for the convection amplitude $A_j$ of rotating RBC in the presence of external periodic forcing (Kelly & Pal Reference Kelly and Pal1978; Coullet Reference Coullet1986; McCoy Reference McCoy2007; Seiden et al. Reference Seiden, Weiss, McCoy, Pesch and Bodenschatz2008):

(3.1)\begin{equation} {\partial}_tA_j = {\varepsilon}A_j + {\xi}_0^2{\nabla}^2A_j - \sum_{i = 1}^ng_0^{ij}|A_i|^2A_j + g_2^j{\delta}_jA_j^{{\ast}m - 1}. \end{equation}

In this model we describe the influence of the imposed bottom texture on the multiple-mode flow pattern by mapping the bottom surface profile to a temperature modulation of the bottom plate (Kelly & Pal Reference Kelly and Pal1978; Seiden et al. Reference Seiden, Weiss, McCoy, Pesch and Bodenschatz2008). The quantity $g_0^{ij}$ is the nonlinear coupling coefficient between the Fourier modes $i$ and $j$ of the flow pattern (Scheel, Mutyaba & Kimmel Reference Scheel, Mutyaba and Kimmel2010); $n$ is the number of dominant modes, i.e. $n = 2$ (3) for the square (hexagonal) pattern; and $\delta _j = c_jh/H$ represents the strength of the external forcing, with the coefficient $c_j = ({1}/{L_xL_y})\int _{ - L_x/2}^{L_x/2}\int _{ - L_y/2}^{L_y/2}2M(x,y) \cos (2{{\rm \pi} }k_x^j{x}+2{{\rm \pi} }k_y^j{y})\, {\rm d}\kern0.06em x\, {\rm d} y$ representing the bottom surface profile (Kelly & Pal Reference Kelly and Pal1978; Seiden et al. Reference Seiden, Weiss, McCoy, Pesch and Bodenschatz2008). We use an area of $60 \times 60$ mm$^2$ $(L_x = L_y = 60$ mm) to evaluate $c_j$, and we obtain $c_j = 0.226$ and 0.172 for the square and hexagonal bottom textures, respectively. The quantity $A_j$ is the normalized complex amplitude, which satisfies the relationship ${\sum _{j=1}^n}{\vert }A_j{\vert }^2 = {(Nu - 1)Ra/Ra}_{c}$ (Cross Reference Cross1980; Cross & Hohenberg Reference Cross and Hohenberg1993), with the Nusselt number ${Nu} = QH/{\lambda }{{\rm \Delta} }T$ representing the global heat transport. Here $Q$ is the heat flux through the fluid layer and ${\lambda }$ is the thermal conductivity of the fluid. The quantity $A_j^{\ast }$ is the complex conjugate of $A_j$. The integer $m$ denotes the degree of resonance, and we consider here resonant forcing ($k_{f}\,{\approx }\,k_0$ and $m = 1$). The coefficients $\xi _0$ and $g_2^j$ represent the spatial variation of $A_j$ and the degree of imperfection, which depend on $\boldsymbol {k}_{f}$ and $\varOmega$.

In view of the symmetry of the flow patterns shown in figures 2b and 2c, one may reconstruct the square pattern with two plane waves with their characteristic wave vectors ($\boldsymbol {k}_1,\boldsymbol {k}_2$) perpendicular to each other (figure 2e), and the hexagonal pattern using three wave vectors ($\boldsymbol {k}_1, \boldsymbol {k}_2, \boldsymbol {k}_3$) equally spaced (figure 2f). Since these convection modes are stationary, with the amplitude of each Fourier mode being equal to that of the others, we have $A_i = A_j$, and the measured amplitude is the superposition of the convection amplitudes in all modes: $A = {\sum _{i = 1}^n}A_i = nA_i$. The coupling coefficient $g_0^{ij} = g_0$ and the imperfection coefficient $g_2^j = g_2$ are set to constants for all modes $(i, j)$.

We consider here a stationary solution for the near-onset flow regime. For each modulation mode the spatial variation of $A$ is negligible since the flow field is nearly periodic (figures 2b and 2c). Accounting for a shift $\varepsilon _0$ of the onset owing to a local increase of the temperature gradient over the bottom texture (McCoy Reference McCoy2007; Seiden et al. Reference Seiden, Weiss, McCoy, Pesch and Bodenschatz2008) and for each imposed pattern $\boldsymbol {k}_{f}\,{\ne }\,\boldsymbol {k}_0$ (Cross Reference Cross1980), we arrive at an amplitude equation for the forced cells: ${(\varepsilon + \varepsilon _0)}A - g_0{\vert }A{\vert }^2A/n + g_2{\delta } = 0$, with $\delta = \sum _{j = 1}^n{\delta }_j$. When the external forcing is absent, $\delta = 0$ and $\boldsymbol {k}_{f} = \boldsymbol {k}_0$. The amplitude equation for the reference cell is thus reduced to ${\varepsilon }A - g_0^rA^3 = 0$, with $g_0^r$ being the nonlinear coupling constant for the reference cell.

In rapidly rotating RBC, the columnar vortices possess similar spatial profiles of temperature, vertical velocity and vertical vorticity (Portegies et al. Reference Portegies, Kunnen, van Heijst and Molenaar2008; Grooms et al. Reference Grooms, Julien, Weiss and Knobloch2010). The amplitude $A$ is thus related to the mean vertical vorticity modulus ${\langle }{\vert }{\omega }{\vert }{\rangle }$ through a scale factor $S$, $A = S{\langle }{\vert }{\omega }{\vert }{\rangle }$, yielding the following bifurcation equations for ${\langle }{\vert }{\omega }{\vert }{\rangle }$:

(3.2)\begin{equation} (\varepsilon + {\varepsilon}_0^s){\langle}{\vert}{\omega}{\vert}{\rangle} - g_0^sS^2{\langle}{\vert}{\omega}{\vert}{\rangle}^3/2 + g_2^s{\delta}^s/S = 0 \end{equation}

for the square-patterned cell,

(3.3)\begin{equation} (\varepsilon + {\varepsilon}_0^h){\langle}{\vert}{\omega}{\vert}{\rangle} - g_0^hS^2{\langle}{\vert}{\omega}{\vert}{\rangle}^3/3 + g_2^h{\delta}^h/S = 0 \end{equation}

for the hexagon-patterned cell, and

(3.4)\begin{equation} {\varepsilon}{\langle}{\vert}{\omega}{\vert}{\rangle} - g_0^rS^2{\langle}{\vert}{\omega}{\vert}{\rangle}^3 = 0\end{equation}

for the reference cell. Here the superscripts $s,h$ denote coefficients for the square and hexagonal patterns, respectively. We fitted the experimental data for ${\langle }{\vert }{\omega }{\vert }{\rangle }(\varepsilon )$ to the theoretical predictions of (3.2)–(3.4) for the forced cells and the reference cell, respectively. In figure 4, both the experimental data and the theoretical curve show clearly the signature of a forward bifurcation near $\varepsilon = 0$ in the reference cell when the external forcing is absent. Meanwhile the pronounced rounding of the transition in the two forced cells suggests an imperfect bifurcation, for which we find the values of the bifurcation terms $g_2^s{\delta }^sS^{-1} = 4.38 \times 10^{-4}(s^{-1})$ and ${g_2^h{\delta }^hS^{-1} = 5.00 \times 10^{-4}}(s^{-1})$. The offsets of the convection onset are found to be $\varepsilon _0^s = 1.0 \times 10^{-3}$ and $\varepsilon _0^h = 1.26 \times 10^{-2}$, signifying that the bottom textures increase the local temperature gradient and destabilize the convection system. Moreover we obtain the coefficient of the coupling terms as $g_0S^2 = (2.60 \times 10^2, 4.16 \times 10^2, 6.81 \times 10^2)(s^2)$ for the reference cell, square-patterned cell and hexagon-patterned cell, respectively.

We consider a dimensionless bifurcation parameter, $G = \sqrt {g_0}g_2$, which reveals the transitional property of the bifurcation curve ${\langle }{\vert }{\omega }{\vert }{\rangle }(\varepsilon )$ near onset. For the two forced cells we find $G^s = \sqrt {g_0^s}g_2^s = 0.416$, $G^h = \sqrt {g_0^h}g_2^h = 0.530$; both are independent of the scale factor $S$. Although there exists to date no complete theory of near-onset bifurcation for rotating convection with external forcing, the amplitude equation of RBC subject to a spatially periodic modulation has been formulated (Kelly & Pal Reference Kelly and Pal1978; McCoy Reference McCoy2007); these papers show that the external forcing results in an imperfection term with a coefficient $g_2^{\ast } = 0.144$, and a coupling coefficient in the cubic term $g_0^{\ast } = 13.05$, in their chosen units. Therefore, the theoretically predicted value of the bifurcation parameter for non-rotating convection, $G^{\ast } = \sqrt {g_0^{\ast }}g_2^{\ast } = 0.520$, appears close to our results for $G^s$ and $G^h$ for the square- and hexagon-patterned cells in rotating convection. Lastly, the agreement between the experimental and theoretical results shown in figure 4 suggests that the physics of external modulation near the onset of rotating convection is well described by the present Ginzburg–Landau-like model.